MinT - Best Known (47−29, 47, s)-Nets in Base 128

Best Known (47−29, 47, s)-Nets in Base 128

(47−29, 47, 342)-Net over F₁₂₈ — Constructive and digital

Digital (18, 47, 342)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (3, 32, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(47−29, 47, 387)-Net over F₁₂₈ — Digital

Digital (18, 47, 387)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁴⁷, 387, F₁₂₈, 3, 29) (dual of [(387, 3), 1114, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(128⁴⁴, 386, F₁₂₈, 3, 29) (dual of [(386, 3), 1114, 30]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(3;F,1128P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(47−29, 47, 513)-Net in Base 128

(18, 47, 513)-net in base 128, using

t-expansion [i] based on (17, 47, 513)-net in base 128, using
- 25 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(47−29, 47, 399334)-Net in Base 128 — Upper bound on s

There is no (18, 47, 399335)-net in base 128, because

1 times m-reduction [i] would yield (18, 46, 399335)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 8 544126 070468 464855 292922 062528 588432 400577 107428 524603 350175 529422 988039 835137 872893 980284 600396 > 128⁴⁶ [i]